Problem: $ B = \left[\begin{array}{rrr}2 & 4 & 0 \\ 5 & 0 & 3\end{array}\right]$ $ D = \left[\begin{array}{rr}-1 & 1 \\ 1 & -1 \\ 2 & 2\end{array}\right]$ What is $ B D$ ?
Solution: Because $ B$ has dimensions $(2\times3)$ and $ D$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ B D = \left[\begin{array}{rrr}{2} & {4} & {0} \\ {5} & {0} & {3}\end{array}\right] \left[\begin{array}{rr}{-1} & \color{#DF0030}{1} \\ {1} & \color{#DF0030}{-1} \\ {2} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ B$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ B$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ B$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{4}\cdot{1}+{0}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ B$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{4}\cdot{1}+{0}\cdot{2} & ? \\ {5}\cdot{-1}+{0}\cdot{1}+{3}\cdot{2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ B$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-1}+{4}\cdot{1}+{0}\cdot{2} & {2}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{-1}+{0}\cdot\color{#DF0030}{2} \\ {5}\cdot{-1}+{0}\cdot{1}+{3}\cdot{2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{-1}+{4}\cdot{1}+{0}\cdot{2} & {2}\cdot\color{#DF0030}{1}+{4}\cdot\color{#DF0030}{-1}+{0}\cdot\color{#DF0030}{2} \\ {5}\cdot{-1}+{0}\cdot{1}+{3}\cdot{2} & {5}\cdot\color{#DF0030}{1}+{0}\cdot\color{#DF0030}{-1}+{3}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}2 & -2 \\ 1 & 11\end{array}\right] $